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When I try to derive the combined Couette - poiseuille flow using non-dimensionalized fundamental equations of compressible viscous flow, I get pressure gradient as a function of both x and y

$ Non-dimensionalized$ $Navier-Strokes$ $Equation $

in x-direction

$$ \frac{\partial u^*}{\partial t^*} + u^*\frac{\partial u^*}{\partial x^*}+v^*\frac{\partial u^*}{\partial y^*} = -\frac{\partial p^*}{\partial x^*}+\frac{1}{Re}(\frac{\partial^{2} u^*}{\partial {x^*}^{2}}+\frac{\partial^{2} u^*}{\partial {y^*}^{2}})+\frac{1}{{Fr}^{2}}\ $$

in y-direction

$$ \frac{\partial v^*}{\partial t^*} + u^*\frac{\partial v^*}{\partial x^*}+v^*\frac{\partial v^*}{\partial y^*} = -\frac{\partial p^*}{\partial y^*}+\frac{1}{Re}(\frac{\partial^{2} v^*}{\partial {x^*}^{2}}+\frac{\partial^{2} v^*}{\partial {y^*}^{2}})+\frac{1}{{Fr}^{2}}\ $$

Where $Re$ is the Reynolds number, and $Fr$ is the Froude number.

Now from our assumptions that the flow is

  1. steady state
  2. fully developed( variation of velocity in the x direction is zero)
  3. incompressible
  4. Newtonian fluid

the above equations can be re-written as

in x- direction

$$-\frac{\partial p^*}{\partial x^*}+ \frac{1}{Re}(\frac{\partial^{2} u^*}{\partial {y^*}^{2}}) + \frac{1}{{Fr}^{2}}\ = 0 $$

in y- direction

$$-\frac{\partial p^*}{\partial y^*}+ \frac{1}{{Fr}^{2}}\ = 0 $$

Which results in the pressure gradient being a function of both x and y, which is not true since, from the poiseuille flow, we know that the pressure gradient is only a function of x

Please explain how can I derive the combined flow using non dimensionalized equations

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1 Answer 1

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My reputation isn't high enough to write this as a comment. However, often one assumes in flows such as Couette, Poiseuille, or Couette-Poiseuille that the fluid isn't being acting under gravity. In such a case, the reciprocal of the Froude number is zero and hence $-\frac{\partial p^*}{\partial y}=0$, so that the pressure gradient is only a function of $x$.

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